Random Errors Always Present''

1
Random Errors Always Present..

• Necessary to analyze and obtain estimates of
  their magnitude
• Use statistical analysis! (Hence, the term
  statistical error.)
• Different results are obtained from one
  measurement to the next - does a true value for
  the measurement actually exist?
• One approach - take large number of measurements,
  use average - impractical in routine practice
• How good is the result of a single measurement as
  an estimate of the true value? - I.e. What is the
  uncertainty of the result?
• Need known frequency (or probability)
  distributions..

2
Uncertainty, Probability, and StatisticsPart 2
- Probabilities
3
Probability - the study of the chance that a
particular event or series of events will occur.
4
Definition 1 Mathematical

• P(A) is a number obeying the Kolmogorov axioms

5
Problem with Mathematical definition

• No information is conveyed by P(A)

6
Definition 2 Classical

• The probability P(A) is a property of an object
  that determines how often event A happens.
• It is given by symmetry for equally-likely
  outcomes
• Outcomes not equally-likely are reduced to
  equally-likely ones
• Examples
• Tossing a coin
• P(H)1/2
• Throwing two dice
• P(8)5/36

7
Problems with the classical definition

• When are cases equally likely?
• If you toss two coins, are there 3 possible
  outcomes or 4?
• Can be handled
• How do you handle continuous variables?
• Split the triangle at random
• Cannot be handled

8
Bertrands Paradox

• A jug contains 1 glassful of water and between 1
  and 2 glasses of wine
• Q What is the most probable winewater ratio?
• A Between 1 and 2 ?3/2
• Q What is the most probable waterwine ratio?
• A Between 1/1 and 1/2 ?3/4
• (3/2)?(3/4)-1

9
Definition 3 Frequentist

• The probability P(A) is the large N limit
  fraction of cases in which A is true (taken over
  some ensemble)

The standard (frequentist) definition has an
interesting property (taught at school) - and an
interesting limitation not often taught at school
(Example - toss a coin)
10
Problem (limitation) for the Frequentist
definition

• P(A) depends on A and the ensemble
• Eg count 10 of a group of 30 with beards.
• P(beard)1/3

11
Aside Consequences for Quantum Mechanics
p

• QM calculates probabilities
• Probabilities are not real they depend on the
  process and the ensemble

L
p-
n
L
p0
PDG P(pp-)0.639 , P(np0)0.358
12
Big problem for the Frequentist definition
Some events are unique. Consider It will
probably rain tomorrow. or even There is a
70 probability of rain tomorrow There is only
one tomorrow (Wednesday). There is NO ensemble.
P(rain) is either 0/1 0 or 1/1 1 Strict
frequentists cannot say 'It will probably rain
tomorrow'.
13
Circumventing the problem

• A frequentist can say
• The statement It will rain tomorrow
• has a 70 probability of being true.
• by assembling an ensemble of
• statements and ascertaining that
• 70 are true.
• (E.g. Weather forecasts
• with a verified
• track record)

14
But that doesnt always work

• Rain prediction in unfamiliar territory
• Higgs discovery
• Dark matter searches

15
(No Transcript)
16
Definition 4 Subjective (Bayesian)

• P(A) is your degree of belief in A
• You will accept a bet on A if the odds are better
  than
• 1-P to P
• A can be Anything
• Beards, Rain, particle decays, conjectures,
  theories

17
Bayes Theorem Often used for subjective
probability

• Conditional Probability P(AB)
• P(A B) P(B) P(AB)
• P(A B) P(A) P(BA)

• Example
• Wwhite jacket Bbald
• P(WB)(2/4)x(1/2)
• or (1/4)x(1/1)
• P(WB)
• 1 x (1/4) 1/2
• (2/4)

18
Conclusion What is Probability?

• 4 ways to define it
• Mathematical
• Classical
• Frequentist
• Subjective
• Each has strong points and weak points
• None is universally applicable
• Be prepared to understand and use them all -